3.903 \(\int \frac{\sec ^5(c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=192 \[ -\frac{\sec ^{10}(c+d x)}{10 a d}+\frac{\sec ^8(c+d x)}{4 a d}-\frac{\sec ^6(c+d x)}{6 a d}+\frac{3 \tanh ^{-1}(\sin (c+d x))}{256 a d}+\frac{\tan ^3(c+d x) \sec ^7(c+d x)}{10 a d}-\frac{3 \tan (c+d x) \sec ^7(c+d x)}{80 a d}+\frac{\tan (c+d x) \sec ^5(c+d x)}{160 a d}+\frac{\tan (c+d x) \sec ^3(c+d x)}{128 a d}+\frac{3 \tan (c+d x) \sec (c+d x)}{256 a d} \]

[Out]

(3*ArcTanh[Sin[c + d*x]])/(256*a*d) - Sec[c + d*x]^6/(6*a*d) + Sec[c + d*x]^8/(4*a*d) - Sec[c + d*x]^10/(10*a*
d) + (3*Sec[c + d*x]*Tan[c + d*x])/(256*a*d) + (Sec[c + d*x]^3*Tan[c + d*x])/(128*a*d) + (Sec[c + d*x]^5*Tan[c
+ d*x])/(160*a*d) - (3*Sec[c + d*x]^7*Tan[c + d*x])/(80*a*d) + (Sec[c + d*x]^7*Tan[c + d*x]^3)/(10*a*d)

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Rubi [A]  time = 0.25289, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {2835, 2611, 3768, 3770, 2606, 266, 43} \[ -\frac{\sec ^{10}(c+d x)}{10 a d}+\frac{\sec ^8(c+d x)}{4 a d}-\frac{\sec ^6(c+d x)}{6 a d}+\frac{3 \tanh ^{-1}(\sin (c+d x))}{256 a d}+\frac{\tan ^3(c+d x) \sec ^7(c+d x)}{10 a d}-\frac{3 \tan (c+d x) \sec ^7(c+d x)}{80 a d}+\frac{\tan (c+d x) \sec ^5(c+d x)}{160 a d}+\frac{\tan (c+d x) \sec ^3(c+d x)}{128 a d}+\frac{3 \tan (c+d x) \sec (c+d x)}{256 a d} \]

Antiderivative was successfully verified.

[In]

Int[(Sec[c + d*x]^5*Tan[c + d*x]^4)/(a + a*Sin[c + d*x]),x]

[Out]

(3*ArcTanh[Sin[c + d*x]])/(256*a*d) - Sec[c + d*x]^6/(6*a*d) + Sec[c + d*x]^8/(4*a*d) - Sec[c + d*x]^10/(10*a*
d) + (3*Sec[c + d*x]*Tan[c + d*x])/(256*a*d) + (Sec[c + d*x]^3*Tan[c + d*x])/(128*a*d) + (Sec[c + d*x]^5*Tan[c
+ d*x])/(160*a*d) - (3*Sec[c + d*x]^7*Tan[c + d*x])/(80*a*d) + (Sec[c + d*x]^7*Tan[c + d*x]^3)/(10*a*d)

Rule 2835

Int[(cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]
), x_Symbol] :> Dist[1/a, Int[Cos[e + f*x]^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[1/(b*d), Int[Cos[e + f*x]
^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2
- b^2, 0] && IntegerQ[n] && (LtQ[0, n, (p + 1)/2] || (LeQ[p, -n] && LtQ[-n, 2*p - 3]) || (GtQ[n, 0] && LeQ[n,
-p]))

Rule 2611

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a*Sec[e
+ f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*(m + n - 1)), x] - Dist[(b^2*(n - 1))/(m + n - 1), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\sec ^5(c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\int \sec ^7(c+d x) \tan ^4(c+d x) \, dx}{a}-\frac{\int \sec ^6(c+d x) \tan ^5(c+d x) \, dx}{a}\\ &=\frac{\sec ^7(c+d x) \tan ^3(c+d x)}{10 a d}-\frac{3 \int \sec ^7(c+d x) \tan ^2(c+d x) \, dx}{10 a}-\frac{\operatorname{Subst}\left (\int x^5 \left (-1+x^2\right )^2 \, dx,x,\sec (c+d x)\right )}{a d}\\ &=-\frac{3 \sec ^7(c+d x) \tan (c+d x)}{80 a d}+\frac{\sec ^7(c+d x) \tan ^3(c+d x)}{10 a d}+\frac{3 \int \sec ^7(c+d x) \, dx}{80 a}-\frac{\operatorname{Subst}\left (\int (-1+x)^2 x^2 \, dx,x,\sec ^2(c+d x)\right )}{2 a d}\\ &=\frac{\sec ^5(c+d x) \tan (c+d x)}{160 a d}-\frac{3 \sec ^7(c+d x) \tan (c+d x)}{80 a d}+\frac{\sec ^7(c+d x) \tan ^3(c+d x)}{10 a d}+\frac{\int \sec ^5(c+d x) \, dx}{32 a}-\frac{\operatorname{Subst}\left (\int \left (x^2-2 x^3+x^4\right ) \, dx,x,\sec ^2(c+d x)\right )}{2 a d}\\ &=-\frac{\sec ^6(c+d x)}{6 a d}+\frac{\sec ^8(c+d x)}{4 a d}-\frac{\sec ^{10}(c+d x)}{10 a d}+\frac{\sec ^3(c+d x) \tan (c+d x)}{128 a d}+\frac{\sec ^5(c+d x) \tan (c+d x)}{160 a d}-\frac{3 \sec ^7(c+d x) \tan (c+d x)}{80 a d}+\frac{\sec ^7(c+d x) \tan ^3(c+d x)}{10 a d}+\frac{3 \int \sec ^3(c+d x) \, dx}{128 a}\\ &=-\frac{\sec ^6(c+d x)}{6 a d}+\frac{\sec ^8(c+d x)}{4 a d}-\frac{\sec ^{10}(c+d x)}{10 a d}+\frac{3 \sec (c+d x) \tan (c+d x)}{256 a d}+\frac{\sec ^3(c+d x) \tan (c+d x)}{128 a d}+\frac{\sec ^5(c+d x) \tan (c+d x)}{160 a d}-\frac{3 \sec ^7(c+d x) \tan (c+d x)}{80 a d}+\frac{\sec ^7(c+d x) \tan ^3(c+d x)}{10 a d}+\frac{3 \int \sec (c+d x) \, dx}{256 a}\\ &=\frac{3 \tanh ^{-1}(\sin (c+d x))}{256 a d}-\frac{\sec ^6(c+d x)}{6 a d}+\frac{\sec ^8(c+d x)}{4 a d}-\frac{\sec ^{10}(c+d x)}{10 a d}+\frac{3 \sec (c+d x) \tan (c+d x)}{256 a d}+\frac{\sec ^3(c+d x) \tan (c+d x)}{128 a d}+\frac{\sec ^5(c+d x) \tan (c+d x)}{160 a d}-\frac{3 \sec ^7(c+d x) \tan (c+d x)}{80 a d}+\frac{\sec ^7(c+d x) \tan ^3(c+d x)}{10 a d}\\ \end{align*}

Mathematica [A]  time = 5.72551, size = 116, normalized size = 0.6 \[ \frac{-\frac{90}{\sin (c+d x)+1}-\frac{45}{(\sin (c+d x)-1)^2}-\frac{45}{(\sin (c+d x)+1)^2}+\frac{40}{(\sin (c+d x)-1)^3}+\frac{20}{(\sin (c+d x)+1)^3}+\frac{30}{(\sin (c+d x)-1)^4}+\frac{90}{(\sin (c+d x)+1)^4}-\frac{48}{(\sin (c+d x)+1)^5}+90 \tanh ^{-1}(\sin (c+d x))}{7680 a d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sec[c + d*x]^5*Tan[c + d*x]^4)/(a + a*Sin[c + d*x]),x]

[Out]

(90*ArcTanh[Sin[c + d*x]] + 30/(-1 + Sin[c + d*x])^4 + 40/(-1 + Sin[c + d*x])^3 - 45/(-1 + Sin[c + d*x])^2 - 4
8/(1 + Sin[c + d*x])^5 + 90/(1 + Sin[c + d*x])^4 + 20/(1 + Sin[c + d*x])^3 - 45/(1 + Sin[c + d*x])^2 - 90/(1 +
Sin[c + d*x]))/(7680*a*d)

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Maple [A]  time = 0.087, size = 180, normalized size = 0.9 \begin{align*}{\frac{1}{256\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{4}}}+{\frac{1}{192\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{3}}}-{\frac{3}{512\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{2}}}-{\frac{3\,\ln \left ( \sin \left ( dx+c \right ) -1 \right ) }{512\,da}}-{\frac{1}{160\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{5}}}+{\frac{3}{256\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{1}{384\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{3}{512\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{3}{256\,da \left ( 1+\sin \left ( dx+c \right ) \right ) }}+{\frac{3\,\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{512\,da}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^9*sin(d*x+c)^4/(a+a*sin(d*x+c)),x)

[Out]

1/256/d/a/(sin(d*x+c)-1)^4+1/192/d/a/(sin(d*x+c)-1)^3-3/512/d/a/(sin(d*x+c)-1)^2-3/512/a/d*ln(sin(d*x+c)-1)-1/
160/d/a/(1+sin(d*x+c))^5+3/256/d/a/(1+sin(d*x+c))^4+1/384/d/a/(1+sin(d*x+c))^3-3/512/a/d/(1+sin(d*x+c))^2-3/25
6/a/d/(1+sin(d*x+c))+3/512*ln(1+sin(d*x+c))/a/d

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Maxima [A]  time = 1.0931, size = 289, normalized size = 1.51 \begin{align*} -\frac{\frac{2 \,{\left (45 \, \sin \left (d x + c\right )^{8} + 45 \, \sin \left (d x + c\right )^{7} - 165 \, \sin \left (d x + c\right )^{6} - 165 \, \sin \left (d x + c\right )^{5} + 219 \, \sin \left (d x + c\right )^{4} - 421 \, \sin \left (d x + c\right )^{3} - 211 \, \sin \left (d x + c\right )^{2} + 109 \, \sin \left (d x + c\right ) + 64\right )}}{a \sin \left (d x + c\right )^{9} + a \sin \left (d x + c\right )^{8} - 4 \, a \sin \left (d x + c\right )^{7} - 4 \, a \sin \left (d x + c\right )^{6} + 6 \, a \sin \left (d x + c\right )^{5} + 6 \, a \sin \left (d x + c\right )^{4} - 4 \, a \sin \left (d x + c\right )^{3} - 4 \, a \sin \left (d x + c\right )^{2} + a \sin \left (d x + c\right ) + a} - \frac{45 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac{45 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{7680 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^9*sin(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/7680*(2*(45*sin(d*x + c)^8 + 45*sin(d*x + c)^7 - 165*sin(d*x + c)^6 - 165*sin(d*x + c)^5 + 219*sin(d*x + c)
^4 - 421*sin(d*x + c)^3 - 211*sin(d*x + c)^2 + 109*sin(d*x + c) + 64)/(a*sin(d*x + c)^9 + a*sin(d*x + c)^8 - 4
*a*sin(d*x + c)^7 - 4*a*sin(d*x + c)^6 + 6*a*sin(d*x + c)^5 + 6*a*sin(d*x + c)^4 - 4*a*sin(d*x + c)^3 - 4*a*si
n(d*x + c)^2 + a*sin(d*x + c) + a) - 45*log(sin(d*x + c) + 1)/a + 45*log(sin(d*x + c) - 1)/a)/d

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Fricas [A]  time = 2.07818, size = 517, normalized size = 2.69 \begin{align*} -\frac{90 \, \cos \left (d x + c\right )^{8} - 30 \, \cos \left (d x + c\right )^{6} - 12 \, \cos \left (d x + c\right )^{4} + 176 \, \cos \left (d x + c\right )^{2} - 45 \,{\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 45 \,{\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (45 \, \cos \left (d x + c\right )^{6} + 30 \, \cos \left (d x + c\right )^{4} - 616 \, \cos \left (d x + c\right )^{2} + 432\right )} \sin \left (d x + c\right ) - 96}{7680 \,{\left (a d \cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^9*sin(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/7680*(90*cos(d*x + c)^8 - 30*cos(d*x + c)^6 - 12*cos(d*x + c)^4 + 176*cos(d*x + c)^2 - 45*(cos(d*x + c)^8*s
in(d*x + c) + cos(d*x + c)^8)*log(sin(d*x + c) + 1) + 45*(cos(d*x + c)^8*sin(d*x + c) + cos(d*x + c)^8)*log(-s
in(d*x + c) + 1) - 2*(45*cos(d*x + c)^6 + 30*cos(d*x + c)^4 - 616*cos(d*x + c)^2 + 432)*sin(d*x + c) - 96)/(a*
d*cos(d*x + c)^8*sin(d*x + c) + a*d*cos(d*x + c)^8)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**9*sin(d*x+c)**4/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.4095, size = 211, normalized size = 1.1 \begin{align*} \frac{\frac{180 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac{180 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac{5 \,{\left (75 \, \sin \left (d x + c\right )^{4} - 300 \, \sin \left (d x + c\right )^{3} + 414 \, \sin \left (d x + c\right )^{2} - 196 \, \sin \left (d x + c\right ) + 31\right )}}{a{\left (\sin \left (d x + c\right ) - 1\right )}^{4}} - \frac{411 \, \sin \left (d x + c\right )^{5} + 2415 \, \sin \left (d x + c\right )^{4} + 5730 \, \sin \left (d x + c\right )^{3} + 6730 \, \sin \left (d x + c\right )^{2} + 3515 \, \sin \left (d x + c\right ) + 703}{a{\left (\sin \left (d x + c\right ) + 1\right )}^{5}}}{30720 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^9*sin(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/30720*(180*log(abs(sin(d*x + c) + 1))/a - 180*log(abs(sin(d*x + c) - 1))/a + 5*(75*sin(d*x + c)^4 - 300*sin(
d*x + c)^3 + 414*sin(d*x + c)^2 - 196*sin(d*x + c) + 31)/(a*(sin(d*x + c) - 1)^4) - (411*sin(d*x + c)^5 + 2415
*sin(d*x + c)^4 + 5730*sin(d*x + c)^3 + 6730*sin(d*x + c)^2 + 3515*sin(d*x + c) + 703)/(a*(sin(d*x + c) + 1)^5
))/d